= Unit form of Hilbert theorem 90
{title2=$N_{L/K}(x)=1\iff x=\phi(y)/y,\quad y\in\mathcal O_L^\times$}
For an <unramified extension> of non-Archimedean <local fields>, let $\phi$ generate its cyclic <Galois group>. The ordinary <Hilbert theorem 90> supplies $b\in L^\times$ with $x=\phi(b)/b$. A common <uniformizer> $\pi\in K$ gives $y=\pi^{-v_L(b)}b\in\mathcal O_L^\times$, retaining the same quotient because $\phi$ fixes $\pi$. The reverse implication follows by telescoping the <field norm>.
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